Knots and Physics

This book is the fourth edition of the classical text ``Knots and Physics'' by Kauffman, first published in 1991. In this new edition typographical errors have been corrected and the author has added a new chapter on Virtual Knot Theory and Khovanov homology. The book is divided in two parts and a long appendix. Part I is essentially a thorough review of Knot Theory from the combinatorial and algebraic viewpoint. Some of the topics covered by this part are Reidemester's theorem, Jones and Alexander-Conway polynomials and $3$-manifold invariants. This part is plenty of illustrative examples and exercises of how to compute knot polynomials, as well as hundreds of pictures indicating the intuition behind the results that are proved in the text. However, I would not say that this is a book for beginners in the subject of knot invariants, but rather an excellent complement to more standard texts in Knot Theory showing the original approach of the author to the subject. Part I also contains connections of Knot Theory with Theoretical Physics, like Yang-Baxter equations, Feynman's diagrams and Witten's approach to knot invariants using path integrals. Kauffman makes sense of the ideas coming from Theoretical Physics using a combinatorial/algebraic setting, although I find some parts of the text hard to understand. The second part of the book is a miscellany on knots and physics. It covers several topics: Potts model in Statistical Physics, Penrose theory of spin networks, and the implications (sometimes rather speculative) of Knot Theoy in String Theory, DNA and Dynamical Systems. I would say that the main moral of this part of the book is that knot invariants should have an appropriate quantum statistical framework, but this is a theory not yet mathematically developed due to the enormous difficulties to give meaning to the functional integration used in Theoretical Physics. The appendix of the book contains four papers by the author, previously published, and two new articles, the first one explaining Witten's heuristic approach to Knot Theory using the path integral, and the second one introducing the recently developed subject of Virtual Knot Theory and Khovanov homology. This book is one of the very few monographs on the many and fruitful connections of Knot Theory and Theoretical Physics. However, it just focuses on the relationship of Knot Theory with Statistical and Quantum Physics, not covering other important areas where knots arise and that have experienced a remarkable recent progress, like Fluid Mechanics, Plasma Physics and Electromagnetic Theory. I recommend this book to any researcher and PhD student interested in the interplay of Topology and Physics in both directions, i.e. how physical ideas can provide an important insight into the problems of Knot Theory, and how knotted fields arise in Theoretical Physics, which is benefited from the tools of Knot Theory.

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